tricky summation proof by induction

I need to prove the following identity by induction:

$$\sum_{i=0}^ni\cdot n^{i-1}=(n-1)\cdot 2^n+1$$

I have the base case where $n=1$ and all that, but I'm stuck on how to turn the inductive step into the final solution. I know I need to assume the identity holds for $n-1$, but I'm stuck on how to use that to show it holds true for n.